I am reading parallely two books on analysis, and they have two different definitions of compact set: 1) Subset A of metric space X is called compact, if every open cover of A contains a finite subcover. 2) Subset A of metric space X is called compact, if from every sequence of point in A you can choose convergent sequence with limit in A.
Is there any proof which shows equivalence of these definitions? I don't even imagine where to begin constructing proof.
The first definition is that of compactness while the second definition is for sequential compactness. For a general topological space, the two notions do not coincide, but for metric spaces they do. A proof of the equivalence between the two definitions in this setting can be found in most topology books (for example, Theorem 28.2 of Munkres' Topology, second edition). There are also many websites which contain the relevant information, such as this one.